Do any authors take the sheaf-theoretic viewpoint on multivalued functions and/or indefinite integrals?
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It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.
For example:
The real square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{R}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{R}$ satisfying $forall x in U(f(x)^2 = x).$
The complex square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(f(z)^2 = z).$
The complex logarithm function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(e^{f(z)} = z).$
If $f$ is a continuous real-valued function, we can define $int f(x) dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $mathfrak{f}$ is a sheaf of continuous functions, we can define $int mathfrak{f}(x)dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable function $F : U rightarrow mathbb{R}$ satisfying $F' in mathfrak{f}(U).$
Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$log(z) subseteq int frac{1}{z}dz,$$ etc. where $log$ is viewed as the sheaf-theoretic inverse of $z mapsto e^z$ as described above, and the $subseteq$ in this context really means something like: for all open $U subseteq mathbb{C}$, we have $$(z mapsto log(z))(U) subseteq left(z mapsto int frac{1}{z}dzright)(U).$$
Question. Do any published books or articles take this point of view?
calculus real-analysis complex-analysis sheaf-theory multivalued-functions
asked Jun 9 '18 at 3:51
goblingoblin
37k1159193
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add a comment |
$begingroup$
It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.
For example:
The real square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{R}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{R}$ satisfying $forall x in U(f(x)^2 = x).$
The complex square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(f(z)^2 = z).$
The complex logarithm function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(e^{f(z)} = z).$
If $f$ is a continuous real-valued function, we can define $int f(x) dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $mathfrak{f}$ is a sheaf of continuous functions, we can define $int mathfrak{f}(x)dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable function $F : U rightarrow mathbb{R}$ satisfying $F' in mathfrak{f}(U).$
Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$log(z) subseteq int frac{1}{z}dz,$$ etc. where $log$ is viewed as the sheaf-theoretic inverse of $z mapsto e^z$ as described above, and the $subseteq$ in this context really means something like: for all open $U subseteq mathbb{C}$, we have $$(z mapsto log(z))(U) subseteq left(z mapsto int frac{1}{z}dzright)(U).$$
Question. Do any published books or articles take this point of view?
calculus real-analysis complex-analysis sheaf-theory multivalued-functions
asked Jun 9 '18 at 3:51
goblingoblin
37k1159193
$endgroup$
1
$begingroup$
I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
$endgroup$
– Cory Griffith
Jun 29 '18 at 17:49
$begingroup$
@CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
$endgroup$
– goblin
Jul 4 '18 at 15:15
add a comment |
$begingroup$
It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.
For example:
The real square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{R}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{R}$ satisfying $forall x in U(f(x)^2 = x).$
The complex square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(f(z)^2 = z).$
The complex logarithm function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(e^{f(z)} = z).$
If $f$ is a continuous real-valued function, we can define $int f(x) dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $mathfrak{f}$ is a sheaf of continuous functions, we can define $int mathfrak{f}(x)dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable function $F : U rightarrow mathbb{R}$ satisfying $F' in mathfrak{f}(U).$
Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$log(z) subseteq int frac{1}{z}dz,$$ etc. where $log$ is viewed as the sheaf-theoretic inverse of $z mapsto e^z$ as described above, and the $subseteq$ in this context really means something like: for all open $U subseteq mathbb{C}$, we have $$(z mapsto log(z))(U) subseteq left(z mapsto int frac{1}{z}dzright)(U).$$
Question. Do any published books or articles take this point of view?
calculus real-analysis complex-analysis sheaf-theory multivalued-functions
asked Jun 9 '18 at 3:51
goblingoblin
37k1159193
$endgroup$
It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.
For example:
The real square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{R}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{R}$ satisfying $forall x in U(f(x)^2 = x).$
The complex square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(f(z)^2 = z).$
The complex logarithm function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(e^{f(z)} = z).$
If $f$ is a continuous real-valued function, we can define $int f(x) dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $mathfrak{f}$ is a sheaf of continuous functions, we can define $int mathfrak{f}(x)dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable function $F : U rightarrow mathbb{R}$ satisfying $F' in mathfrak{f}(U).$
Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$log(z) subseteq int frac{1}{z}dz,$$ etc. where $log$ is viewed as the sheaf-theoretic inverse of $z mapsto e^z$ as described above, and the $subseteq$ in this context really means something like: for all open $U subseteq mathbb{C}$, we have $$(z mapsto log(z))(U) subseteq left(z mapsto int frac{1}{z}dzright)(U).$$
Question. Do any published books or articles take this point of view?
calculus real-analysis complex-analysis sheaf-theory multivalued-functions
calculus real-analysis complex-analysis sheaf-theory multivalued-functions
asked Jun 9 '18 at 3:51
goblingoblin
37k1159193
asked Jun 9 '18 at 3:51
goblingoblin
37k1159193
asked Jun 9 '18 at 3:51
goblingoblin
37k1159193
asked Jun 9 '18 at 3:51
goblingoblin
37k1159193
asked Jun 9 '18 at 3:51
goblingoblin
37k1159193
37k1159193
1
$begingroup$
I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
$endgroup$
– Cory Griffith
Jun 29 '18 at 17:49
$begingroup$
@CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
$endgroup$
– goblin
Jul 4 '18 at 15:15
add a comment |
1
$begingroup$
I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
$endgroup$
– Cory Griffith
Jun 29 '18 at 17:49
$begingroup$
@CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
$endgroup$
– goblin
Jul 4 '18 at 15:15
1
1
$begingroup$
I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
$endgroup$
– Cory Griffith
Jun 29 '18 at 17:49
$begingroup$
I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
$endgroup$
– Cory Griffith
Jun 29 '18 at 17:49
$begingroup$
@CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
$endgroup$
– goblin
Jul 4 '18 at 15:15
$begingroup$
@CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
$endgroup$
– goblin
Jul 4 '18 at 15:15
add a comment |
1 Answer
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Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.
answered Dec 15 '18 at 5:26
Alexandre EremenkoAlexandre Eremenko
2,415921
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$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58
add a comment |
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Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.
answered Dec 15 '18 at 5:26
Alexandre EremenkoAlexandre Eremenko
2,415921
$endgroup$
$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58
add a comment |
$begingroup$
Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.
answered Dec 15 '18 at 5:26
Alexandre EremenkoAlexandre Eremenko
2,415921
$endgroup$
$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58
add a comment |
$begingroup$
Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.
answered Dec 15 '18 at 5:26
Alexandre EremenkoAlexandre Eremenko
2,415921
$endgroup$
Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.
answered Dec 15 '18 at 5:26
Alexandre EremenkoAlexandre Eremenko
2,415921
answered Dec 15 '18 at 5:26
Alexandre EremenkoAlexandre Eremenko
2,415921
answered Dec 15 '18 at 5:26
Alexandre EremenkoAlexandre Eremenko
2,415921
answered Dec 15 '18 at 5:26
Alexandre EremenkoAlexandre Eremenko
2,415921
2,415921
$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58
add a comment |
$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58
$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58
$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58
add a comment |
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I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
$endgroup$
– Cory Griffith
Jun 29 '18 at 17:49
$begingroup$
@CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
$endgroup$
– goblin
Jul 4 '18 at 15:15